Analytical Vibration Solutions of Sandwich Lévy Plates with Viscoelastic Layers at Low and High Frequencies
Abstract
1. Introduction
2. Problem Statement
3. Governing Equations
3.1. Single Plate on a Viscoelastic Foundation
3.2. Double-Plate Structure with a Fully Distributed Viscoelastic Layer
3.3. Double-Plate Structure with a Partially Distributed Viscoelastic Layer
3.4. Three-Plate Structure with Dislocated Viscoelastic Layers
4. State-Space Formulation in the s-Domain
4.1. Formulation for Single Plate on a Viscoelastic Foundation
4.2. Formulation for Double-Plate Structure with a Fully Distributed Viscoelastic Layer
- For plate 1
- For plate 2
4.3. Formulation for Double-Plate Structure with a Partially Distributed Viscoelastic Layer
- State equation
- Boundary condition
4.4. Formulation for Three-Plate Structure with Dislocated Viscoelastic Layers
4.5. Discussion
5. Models of Viscoelastic Layers
5.1. Kelvin–Voigt Model
5.2. Maxwell Model
5.3. Standard Linear Solid Models
6. Vibration Analysis by the DTFM
6.1. Plate Structures with Fully Distributed Viscoelastic Layers
- State equation
- Boundary condition
6.2. Plate Structures with Partially Distributed Viscoelastic Layers
6.3. Discussion
7. Numerical Examples
7.1. Example 1: A Double-Plate Structure with a Fully Connected Viscoelastic Layer
7.2. Example 2. A Double-Plate Structure with a Partially Distributed Viscoelastic Layer
7.3. Example 3. A Three-Plate Structure with Dislocated Viscoelastic Layers
8. Conclusions
- (1)
- The proposed method, which is developed based on the Distributed Transfer Function Method (DTFM), provides an -domain state-space formulation for multi-layer sandwich plates with various configurations (layouts) of viscoelastic layers. This formulation does not rely on any spatial discretization in modeling. As a result, closed-form analytical solutions to the frequency response problem of sandwich Lévy plates with various configurations are obtained for the first time.
- (2)
- One unique feature of the proposed method is that the model development and solution procedure for frequency responses remain unchanged in all frequency regions. This consistency in modeling and analysis differs from FEA-based methods, which need to implement mesh refinement at different frequencies. Accordingly, the DTFM-based approach provides a unified framework for frequency response analysis of sandwich Lévy plates over a wide frequency range.
- (3)
- The proposed method can effectively handle multi-layer sandwich plates with dislocated viscoelastic layers, and it can easily implement different models of viscoelastic materials. These issues would be extremely difficult to manage by conventional analytical methods, if not impossible.
- (4)
- Another special feature of the proposed method is that detailed local information of a sandwich plate in high-frequency vibrations, including the displacements, bending moments, and shear forces of a plate component at any location, can be conveniently obtained from the state vectors. On the other hand, FEA-based methods would require significant effort to determine shear forces, and energy-based methods have difficulty providing detailed local information about the plate response.
- (5)
- The proposed method is illustrated in three numerical examples, where the accuracy, efficiency, and versatility of the DTFM in computing the frequency responses of sandwich Lévy plates is validated through comparison with the FEM and analytical solutions (Navier solutions). It is shown that the DTFM-based analysis works in all frequency regions, from low to high frequencies. Moreover, the simulations reveal that the computational efforts required by the proposed method are much less than those needed by the existing methods.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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FEM (Elements) | DTFM | |||
---|---|---|---|---|
50 | 200 | 3200 | ||
1 | 85.0390 | 86.4263 | 86.8882 | 86.9195 |
2 | 88.7757 | 90.1055 | 90.5486 | 90.5786 |
3 | 161.1637 | 165.4778 | 167.0437 | 167.1528 |
4 | 163.1664 | 167.4287 | 168.9766 | 169.0845 |
5 | 260.1657 | 265.3827 | 267.3096 | 267.4445 |
10 | 323.0613 | 341.1225 | 348.1126 | 348.6107 |
20 | 599.0271 | 645.5216 | 667.3677 | 669.0969 |
30 | 897.0924 | 983.4272 | 989.3284 | 989.8729 |
40 | 1067.6546 | 1164.9535 | 1204.1859 | 1210.4549 |
50 | 1361.6047 | 1510.5935 | 1546.7057 | 1551.3877 |
FEM (Elements) | DTFM | |||
---|---|---|---|---|
50 | 200 | 3200 | ||
1 | 102.8850 | 105.0855 | 105.8691 | 105.9233 |
2 | 105.9944 | 108.1316 | 108.8933 | 108.9460 |
3 | 207.8806 | 213.1652 | 215.3394 | 215.4956 |
4 | 209.4369 | 214.6833 | 216.8422 | 216.9973 |
5 | 267.9959 | 274.1810 | 276.7941 | 276.9834 |
10 | 369.5161 | 375.1433 | 380.1241 | 380.7771 |
20 | 708.8662 | 731.5818 | 756.1503 | 758.2217 |
30 | 980.7883 | 993.7429 | 1037.4766 | 1042.3999 |
40 | 1270.8757 | 1312.7018 | 1347.0952 | 1351.1562 |
50 | 1600.1592 | 1570.7294 | 1624.9480 | 1634.6596 |
FEM (Elements) | DTFM | |||
---|---|---|---|---|
50 | 200 | 3200 | ||
1 | 103.6084 | 105.8236 | 106.6055 | 106.6595 |
2 | 105.6672 | 107.8404 | 108.6079 | 108.6609 |
3 | 219.8495 | 225.0578 | 227.2117 | 227.3665 |
4 | 220.5723 | 225.7630 | 227.9101 | 228.0644 |
5 | 250.9591 | 257.1126 | 259.6909 | 259.8766 |
10 | 397.4359 | 402.7193 | 406.4568 | 406.7532 |
20 | 713.0100 | 765.3261 | 789.3563 | 791.3876 |
30 | 981.4164 | 988.2476 | 1042.9694 | 1047.7172 |
40 | 1308.6136 | 1313.9311 | 1400.5220 | 1401.4399 |
50 | 1523.5747 | 1575.7178 | 1690.4022 | 1699.8813 |
FEM (Elements) | DTFM | |||
---|---|---|---|---|
300 | 1200 | 4800 | ||
1 | 53.7829 | 53.7028 | 53.6821 | 53.6752 |
2 | 54.9444 | 54.8623 | 54.8412 | 54.8341 |
3 | 57.4559 | 57.3714 | 57.3497 | 57.3425 |
4 | 75.1837 | 75.1723 | 75.1687 | 75.1674 |
5 | 75.5086 | 75.4956 | 75.4916 | 75.4902 |
10 | 217.4733 | 216.8162 | 216.6319 | 216.5686 |
20 | 313.8678 | 316.1529 | 316.7388 | 316.9354 |
30 | 492.1189 | 490.0905 | 489.4404 | 492.6234 |
40 | 657.7398 | 661.2479 | 662.5154 | 662.9800 |
50 | 874.2904 | 872.5249 | 870.9846 | 870.5649 |
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Zhang, Y.; Yang, B. Analytical Vibration Solutions of Sandwich Lévy Plates with Viscoelastic Layers at Low and High Frequencies. Appl. Mech. 2025, 6, 49. https://doi.org/10.3390/applmech6030049
Zhang Y, Yang B. Analytical Vibration Solutions of Sandwich Lévy Plates with Viscoelastic Layers at Low and High Frequencies. Applied Mechanics. 2025; 6(3):49. https://doi.org/10.3390/applmech6030049
Chicago/Turabian StyleZhang, Yichi, and Bingen Yang. 2025. "Analytical Vibration Solutions of Sandwich Lévy Plates with Viscoelastic Layers at Low and High Frequencies" Applied Mechanics 6, no. 3: 49. https://doi.org/10.3390/applmech6030049
APA StyleZhang, Y., & Yang, B. (2025). Analytical Vibration Solutions of Sandwich Lévy Plates with Viscoelastic Layers at Low and High Frequencies. Applied Mechanics, 6(3), 49. https://doi.org/10.3390/applmech6030049